_Conjugate observables_ or _incompatible observables_ are pairs of [observables](Observable.md) in quantum mechanics that do not [commute.](Commutators%20in%20quantum%20mechanics.md) What this means is that one can measure a quantity associated with [state](State%20vector), $|\psi\rangle$ corresponding with the first observable without affecting the measured quantity corresponding to the other observable. The commutation relation for a pair of conjugate observables, $\hat{A}$ and $\hat{B}$ takes the following form: $[\hat{A},\hat{B}]=i\hbar\hat{C}$ %%What is C like and is this true for ALL conjugate pairs? C I believe should commute with A and B and there are examples with angular momentum where it does, but I need to check this.%% %%The only place where I've seen this relation in this exact form is Gerry and Knights quantum optics book on page 150, where the context is quadrature sqeezing.%% %%Note also the relation between this form of the commutation relation and its use in the Groenewold-van Hove theorem. see page 201 of Woit's book.%% # Relationship between conjugate observables and conjugate variables One may notice that these commutation relations are reminiscent of [Poisson brackets](Poisson%20bracket.md) that are formed between [conjugate variables](Conjugate%20variables.md) in classical mechanics. Indeed they may be thought of as the replacement of Poisson brackets in [quantum mechanics.](Quantum%20Mechanics%20(index).md) In fact the relation between Poisson brackets and corresponding commutators goes as $\{A,B\}\rightarrow i\hbar [\hat{A},\hat{B}]$ ^ed9687 As part of the standard [quantization](Quantization.md) procedure where we also promote $A$ and $B$ to observables. # Position-momentum commutators ([... see more](Position-Momentum%20Commutators.md)) # Quantum Uncertainty Relations Pairs of conjugate variables in quantum mechanics also form [Quantum uncertainty relations.](Quantum%20uncertainty%20relations) #QuantumMechanics/QuantumMeasurement/QuantumObservables #QuantumMechanics/MathematicalFoundations